If you read what I write here, you may have formed the impression that there are things about modern education that I am not entirely happy about. Time for a little rant?

In fact this was partly triggered by a comment on Bob Walters’ blog, where he claims that we are producing too many mathematicians. I don’t want to discuss that claim here, but I would like to return to it sometime in the future. I’d like to say a bit about what kind of mathematicians we are producing.

It is fashionable now to talk about “teaching-and-learning” as if it is all one word describing a single concept. Students are supposed to learn by discovery, and the teacher simply “facilitates” this kind of discovery. I’m afraid that I never saw a student “discover” for her/himself any deep mathematical truth over which Archimedes or Gauss struggled. I take a rather different and unfashionable view of the process: I think that teachers teach, and learners learn.

I am not saying that I don’t learn from my students. I have learned an immense amount from them, in many ways: some have discovered entirely new approaches to problems; some have found a body of theory which can be used on a problem of which I was unaware; many have, by their questions, shown me that I didn’t understand something as well as I thought I did, and set me on my own learning path. But I do think there is a difference between this and the bread-and-butter of teaching and learning.

We are now being encouraged to adopt a new computer system into our teaching. This goes by the name of “Blackboard” (which I consider almost a blasphemy, since it is so opposed to the kind of teaching associated with writing on a blackboard). It will be a new way of providing information to students on the web. As far as I can see, it has only two features which can’t be done by a “traditional” course web page:

- You can include a discussion. Ideal if learning is discovery, since the students can help one another discover. But is it?
- You can keep people out; only those invited by the moderator can view the page, and some information is restricted to a single individual. I can see how this appeals to the “intellectual property” side of the institution; but I am proud of the fact that people in many places have used lecture notes and other teaching material that I have posted on the web. I have no desire to keep them out.

I often tell my students that a good mathematician is lazy. This is not just to reassure myself; what I mean is that we look for the simplest possible counterexample, or the simplest possible proof, of an assertion. We can be fiercely concentrated during this search, but there is good reason for looking for the simple, apart from the time saved: we are more likely to find it!

So this is a rather special use of the word “lazy”. When I was a student, I spent a great deal of time doing other things – running, making music, and so on – and scarcely ever found my way into the University library. But two things that I did worked so well that I have no hesitation in recommending them to my own students:

- I concentrated very hard during lectures, and produced a first draft of lecture notes that was as good as I could make it. Rather than simply copying down what was on the board, I included the lecturer’s asides, and my own thoughts, even my own anticipation of how a proof would go.
- To prepare for an exam, I summarised the contents of the course on a single sheet of paper, folded it up, and put it in my shirt pocket. I never took it out to look at it, and claimed that it made its way into my brain by a sort of osmosis. Of course, it was preparing the sheet that achieved the result.

I recognised something similar in Oxford, where the coolest thing was never to be seen doing any work. The best students burnt the midnight oil so that they could seem during the day to be completely casual about their subject.

The point is that you can’t escape doing the work if you want to understand the subject. Students often write on their questionnaires that they cannot understand what they are told in lectures. They are asking for more spoon-feeding, and if they don’t get it, then (as consumers) they feel dissatisfied with the educational process; but if we didn’t withhold it, we would be doing them a grave disservice.

It is a truism that I am almost embarrassed to mention, but no less true for that: when students leave university and take a job, they will not be spoon-fed any more. They will be faced with problems to which the answer is not obvious, and expected to do something. If we have not trained them to do this, we have not done our job. I also think that, if teachers explain this to students, they will understand it and will be supportive of our efforts.

I actually believe that a mathematics education can be a very good training for the “real world”. But that is another topic, I think.

I agree with your sentiments. As someone who did a maths a degree more than 30 years ago, and who has had a job in the ‘real world’ ever since, I think a maths education should be about (i) discovering the beauty of the subject and (ii) having the ability (and the confidence that comes with that) to think in a certain ‘mathematical’ way.

Now with a young son at school, I just let my conversations with him about maths go in whatever direction we want for a while, spotting patterns etc, but he usually then wants to know the ‘proof’.

I think a teacher should (probably in this order): excite the student with the ‘big picture’; guide the student through how to prove things; encourage the student to explore themselves; and, then, make sure they practise, and become adept with the concepts.